We prove spectral multiplier theorems for Hormander classes \(\mathcal {H}^\alpha _p\) for 0-sectorial operators A on Banach spaces assuming a bounded \(H^\infty (\Sigma _\sigma )\) calculus for some \(\sigma \in (0,\pi )\) and norm and certain R-bounds on one of the following families of operators: the semigroup \(e^{-zA}\) on \(\mathbb {C}_+,\) the wave operators \(e^{isA}\) for \(s \in \mathbb {R},\) the resolvent \((\lambda - A)^{-1}\) on \(\mathbb {C}\backslash \mathbb {R},\) the imaginary powers \(A^{it}\) for \(t \in \mathbb {R}\) or the Bochner–Riesz means \((1 - A/u)_+^\alpha \) for \(u > 0.\) In contrast to the existing literature we neither assume that A operates on an \(L^p\) scale nor that A is self-adjoint on a Hilbert space. Furthermore, we replace (generalized) Gaussian or Poisson bounds and maximal estimates by the weaker notion of R-bounds, which allow for a unified approach to spectral multiplier theorems in a more general setting. In this setting our results are close to being optimal. Moreover, we can give a characterization of the (R-bounded) \(\mathcal {H}^\alpha _1\) calculus in terms of R-boundedness of Bochner–Riesz means.